of the doable refraction in strained glass. 317 



value of \ which makes the expression in square brackets a 

 minimum. Obviously that expression is never actually zero, since 



\ . /27rCaSk 

 sin 



2irCaSk 



is always numerically less than unity. Accordingly a perfectly 

 black band is not to be expected, and the pale character of the 

 band may be taken as a good measure of the want of uniformity 

 in the stress. 



Let us write 



ZirCaT _ Sk_ 

 \ ~ x > T~ V ' 



Then we have to find for what value of x the quantity 



. I ' sin px \ 



■■Ail — cos x 



px J 



is a minimum. 



-=- = 0, when 



x dA\ , . . 



A dx~) ( pX ~ C ° S X Sln ^ 



— px cos px cos x 4- sin px cos x + x sin px sin x = 0. 



Now p will in general be a small quantity, if the stress has 

 been adjusted to be roughly uniform, so that we may suppose 

 cospx and sin px expanded in powers of px. If we then neglect 

 5th powers, we find 



i£{ l -«« (l-^)[ + eos,(f) + sin,(l-f) = 0. 



If p were zero, the minima (and these are the only solutions of 

 the above equation we are concerned with) would be given by 



x = 2mr, 

 n being an integer. 



Let us assume x = 2nir — y, 

 1 dA „ . . /_ p 2 x 2 \ (p-x p 2 x 2 1 dA\ 



y is here small. If we neglect its square, we find, for a first 

 approximation 



p*x /, x dA 



3 V iA dx 



